Indeed, if \(E\)  denotes the event that \(\Lambda _{3R}\)  contains both an open circuit and
a dual open circuit surrounding \(\Lambda _{2R}\) , with the open one being connected to \(\partial \Lambda _{4R}\) ,
then it is shown in [1]} that
\(\pi _4(p,r,R)\ge \phi _{p}[E] \inf _{\mathcal {D}}\phi _{\mathcal {D},p}[ \mathbf {M}_r(\mathcal {D},R)],\) 
